“The p-chain spectral sequence”

“The p-chain spectral sequence”

by

James F. Davis and Wolfgang L¨ uck

Abstract

We introduce a new spectral sequence called thep-chain spectral se- quence which converges to the (co-)homology of a contravariantC-space with coefficients in a covariant C-spectrum for a small category C. It is different from the corresponding Atiyah-Hirzebruch type spectral se- quence. It can be used in combination with the Isomorphism Conjectures of Baum-Connes and Farrell-Jones to compute algebraicK- andL-groups of group rings and topologicalK-groups of reduced groupC^∗-algebras.

Key words: spaces and spectra over a category,p-chain spectral sequence, K- andL-groups of group rings and groupC^∗-algebras.

Mathematics subject classification 2000: 55T99, 55N99, 19B28, 19D50, 19G24, 19K99, 57R67

0 Introduction

In [9] we defined abelian groupsH_n^C(X;E) whenX:C →SPACESis a con- travariant functor andE:C →SPECTRAis a covariant functor¹. These abelian groups behave like a generalized homology theory defined on C-spaces, for ex- ample, there is a long exact Mayer-Vietoris sequence. They are weak homotopy invariant; given a mapf:X →Y of C-spaces (i.e. a natural transformation) which induces a weak homotopy equivalencef(c) :X(c)→Y(c) for all objects c∈ObC, thenf_∗:H_n^C(X;E)→H_n^C(Y;E) is an isomorphism.

Three special cases will illustrate these groups:

(a) Fixing an object c ∈ ObC, H_n^C(mor_C(?, c)) = π_n(E(c)). This should be thought of as giving the coefficients of the generalized homology theory;

(b) IfX =? is the constant functor, thenH_n^C(?;E) =π_n(hocolim_CE);

(c) IfC is a category with a single object, all of whose morphisms are isomor- phisms, our generalized homology theory reduces to Borel homology. More precisely, let Gbe the group of morphisms and letX be a CW-complex with an action ofGby cellular maps, then

H_n^C(X;E) =H_n^G(X;E) =:πn((X×EG)+∧GE).

1We also defined cohomology groupsH_Cⁿ(X;E) when X:C → SPACES and E:C → SPECTRAarebothcontravariant functors. In this introduction we will only discuss homology in order to simplify the exposition.

In [9] we gave a spectral sequence converging toH_p+q^C (X;E) whoseE²-term isE_p,q² =H_p^C(X;πq(E)). This spectral sequence is both quite useful and quite standard. It is related to spectral sequences of Atiyah-Hirzebruch, Bousfield- Kan, and Quinn. The point of the current paper is to introduce anewspectral sequence. It converges toH_p+q^C (X;E) and has asE¹-term

E_p,q¹ = a

c_∗∈Q_p

i=0Is(C), S(c_∗)6=∅

H_q^C(X(c_p)×aut(c_p)S(c_∗)×aut(c₀)mor(?, c₀);E),

where Is(C) is the set of isomorphism classes of objects inCandS(c_∗) is a certain aut(cp)-aut(c0)-set. In the special case of an orbit category Or(G,F), whereG is a (discrete) group and F is a family of subgroups satisfying the condition gHg⁻¹⊆H⇒gHg⁻¹=H forg∈GandH ∈ F, thenE_p,q¹ can be indexed by

“p-chains”

(H₀)<(H₁)< . . . <(H_p)

where (H_i) is the conjugacy class of H_i ∈ F and (H_i) < (H_i+1) means that there exists g ∈ G so that gH_ig⁻¹ $ H_i+1. More generally, if C is a EI- category (endomorphisms are isomorphisms), then E_p,q¹ is indexed by similar p-chains. TheE_p,q¹ -term looks formidable, but we discuss many simplifications and reinterpretations in cases of interest. In general, theE¹-term of the spectral sequence is related to Borel homology of the groups {aut(c)}c∈ObC and the differentials are related to assembly maps.

It should be emphasized that the motivation for this abstract-looking spec- tral sequence comes from geometric topology (the surgery classification of man- ifolds), geometry (which manifolds admits metrics of positive scalar curvature), and analysis (the study ofK-theory of groupC^∗-algebras), in conjunction with the study of (fundamental) groups which are infinite, but contain torsion. The connection with these various subjects comes through isomorphism conjectures and assembly maps, for more on this see [9] and the end of Section 1 of the current paper.

In Section 1 we review the definition of H_n^C(X;E) and discuss maps con- nected with a functor F: C → D. In Section 2 we derive the p-chain spec- tral sequence and its differentials and discuss simplifications which occur when the categoryC is left-free. In Section 3 we discuss the important special case H_n^C(?;E) where C is the restricted orbit category. Here, by using the Cofinal- ity Theorem and an analogue of Quillen’s Theorem A, computations can be simplified. We also discuss differentials in thep-chain spectral sequence, which often turn out to be assembly maps themselves. In Section 4 we give examples of groups where the methods of the preceeding sections in combination with the Isomorphism Conjectures of Baum-Connes and Farrell-Jones lead to ex- plicit computations of algebraicK- andL-groups of group rings and topological K-groups of reduced group C^∗-algebras.

Thep-chain spectral sequence is a generalization of the spectral sequence in [17, Chapter 17] and is related to the paper of Slominska [29].

The authors thank Bill Dwyer and Reiner Vogt for useful conversations.

The second author thanks the Max-Planck Institut f¨ur Mathematik in Bonn for its hospitality during his stay in November/December 2002 when parts of this paper were written.

The paper is organized as follows:

0. Introduction

1. Review of Spaces over a Category and Assembly maps 2. Thep-Chain Spectral Sequence

3. Assembly Maps 4. Examples

References

1 Review of Spaces over a Category and Assembly Maps

In this section we review some basic facts from [9] for the convenience of the reader.

Let SPACES and SPACES₊ be the categories of topological spaces and pointed topological spaces respectively. We will always work in the category of compactly generated spaces (see [30] and [33, I.4]). Aspectrum

E={(E(n), σ(n))|n∈Z}

is a sequence of pointed spaces {E(n) | n ∈ Z} together with pointed maps σ(n) :E(n)∧S¹→E(n+ 1), calledstructure maps. A(strong) mapof spectra f: E → E⁰ is a sequence of maps f(n) :E(n) → E⁰(n) which are compatible with the structure maps, i.e. f(n+ 1)◦σ(n) = σ⁰(n)◦(f(n)∧idS¹) holds for alln∈Z. This should not be confused with the notion of map of spectra in the stable category (see [1, III.2.]). A spectrum is called an Ω-spectrumif the adjoint of each structure mapE(n)→ΩE(n+ 1) is a weak homotopy equivalence. The homotopy groups of a spectrum are defined by

π_i(E) = colim

n→∞π_i+n(E(n)) where the systemπ_i+n(E(n)) is given by the composite

πi+n(E(n))−→^S πi+n+1(E(n)∧S¹)−−−→^σ(n)∗ πi+n+1(E(n+ 1))

of the suspension homomorphism and the homomorphism induced by the struc- ture map.

Let C be a small category, i.e. a category such that the objects and the morphisms form sets. Acovariant (or contravariant) C-space, pointed C-space,

C-spectrum, . . . is a covariant (or contravariant) functor from C to SPACES, SPACES+,SPECTRA,. . .and a morphism is a natural transformation.

LetX be a contravariant andY be a covariantC-space. Define theirtensor productto be the space

X⊗_CY = a

c∈Ob(C)

X(c)×Y(c)/∼

where ∼ is the equivalence relation which is generated by (xφ, y) ∼ (x, φy) for all morphisms φ: c → d in C and points x ∈ X(d) and y ∈ Y(c). Here xφ stands for X(φ)(x) and φy for Y(φ)(y). If X and Y are C-spaces of the same variance, denote by hom_C(X, Y) the space of maps of C-spaces from X to Y with the subspace topology coming from the obvious inclusion into Q

c∈Ob(C)map(X(c), Y(c)). If Y is a C-space andZ a space, let Y ×Z be the C-space with the same variance asY whose value at the objectcinCisY(c)×Z.

ForC-spacesX andY of the same variance, the set of homotopy classes of maps ofC-spaces [X, Y]^C is defined using maps of C-spacesX×[0,1]→Y. IfY is a C-space andZ a space, let map(Y, Z) be theC-space with the opposite variance asY whose value at the object cin C is map(Y(c), Z). If X is a contravariant C-space, Y a covariant C-space and Z a space, there is a canonical adjunction homeomorphism [9, Lemma 1.5]

map(X⊗CY, Z)−→^∼= hom_C(X,map(Y, Z)).

All of the above notions also make sense for pointed spaces; one has to substitute wedges for disjoint unions, smash products for cartesian products, and pointed mapping spaces for mapping spaces.

A contravariant C-CW-complex X is a contravariant C-space X together with a filtration

∅=X₋1⊆X0⊆X1⊆X2⊆. . .⊆Xn ⊆. . .⊆X = [

n≥0

Xn

such thatX = colimn→∞Xn and for anyn≥0 then-skeleton Xn is obtained from the (n−1)-skeletonXn−1by attachingC-n-cells, i.e. there exists a pushout ofC-spaces of the form

`

i∈I_nmor_C(?, ci)×Sⁿ⁻¹ −−−−→ Xn−1



 y



 y

`

i∈I_nmor_C(?, ci)×Dⁿ −−−−→ Xn

where the vertical maps are inclusions,In is an index set, and theci are objects of C. The definition of a covariant C-CW-complex is analogous. In [9] these were called contravariant free and covariant freeC-CW-complexes, we will omit the word free here. One of the main properties of CW-complexes carries over toC-CW-complexes, namely, a mapf:Y →Z ofC-spaces is a weak homotopy

equivalence, i.e. f(c) is a weak homotopy equivalence of spaces for all objectsc inC, if and only if for anyC-CW-complexX the induced map

f_∗: [X, Y]^C→[X, Z]^C [g]7→[g◦f]

between the homotopy classes of maps ofC-spaces is bijective [9, Theorem 3.4].

In particular Whitehead’s Theorem carries over: a map ofC-CW-complexes is a homotopy equivalence if and only if it is a weak homotopy equivalence. AC-CW- approximation (X, f) of a C-spaceY consists of a C-CW-complexX together with a weak homotopy equivalencef:X →Y. Such a C-CW-approximation always exists, there is even a functorial construction. If (X, f) and (X⁰, f⁰) are twoC-CW-approximations of aC-spaceY, there is a homotopy equivalence h:X → X⁰ ofC-spaces which is determined uniquely up to homotopy by the property thatf⁰◦handf are homotopic.

Let (X, A) be a pair of contravariant pointedC-spaces. Denote the reduced cone of the pointed spaceA by cone(A). For a covariant C-spectrum Edefine

E^C_q(X, A) = πq(X∪Acone(A)⊗CE).

Given a contravariantC-spectrumE, define

E^q_C(X, A) = π_−q(hom_C(X∪Acone(A),E)).

Define (X₊, A₊) to be the pair of contravariant pointed C-spaces which is ob- tained from (X, A) by adding a disjoint base point. Let (u, v) : (X⁰, A⁰) → (X, A) be a C-CW-approximation. For a covariant C-spectrum E define the homology of(X, A)with coefficients inEby

H_q^C(X, A;E) = E^C_q(X₊⁰ , A⁰₊).

Given a contravariant C-spectrum E, define the cohomology of (X, A) with coefficients inEby

H_C^q(X, A;E) = E^q_C(X₊⁰, A⁰₊).

WhenAis empty we omit it from the notation.

ThenH_q^C(X, A;E) andH_C^q(X, A;E) are unreduced homology and cohomol- ogy theories on pairs of C-spaces which satisfy the WHE-axiom, which says that a weak homotopy equivalence induces an isomorphism on (co-)homology.

The homology theory satisfies the disjoint union axiom. The cohomology the- ory satisfies the disjoint union axiom provided thatE is a C-Ω-spectrum. Let

?_C, or briefly ?, be the C-space which takes each object to a point. Then for (X, A) = (?,∅) then the above notions reduce to

πq(hocolim

C E)

and

π₋q(holim

C E)

respectively. If C is the category associated to a group G, i.e. C has a single object, the morphisms inC are in one-to-one correspondence with G, and the composition law inCcorresponds to multiplication inG, thenH_q^C(X, A;E) can be identified with Borel homologyπq(EG×(X+, A+)∧GE), and similarly for co- homology, provided that (X, A) is aCW-pair with aG-action by cellular maps.

Finally, if C is the trivial category with precisely one object and morphism, these notions reduce to the standard notions of the homology and cohomology of spaces given by a spectrum.

Given a functorF:C → Dand aD-space (or spectrum)Y, define therestric- tion ofY with respect toF to be theC-space (or spectrum)F^∗Y(c) =Y(F(c)).

For X a contravariant, respectively covariant, C-space define the induction of X with respect toF to be the D-space

F_∗X(??) =X(?)⊗_Cmor_D(??, F(?)), respectively

F_∗X(??) = mor_D(F(?),??)⊗_C X(?).

There are natural adjunction isomorphisms (see [9, Lemma 1.9])

X⊗_CF^∗Y ∼= F_∗X⊗_DY, (1.1) hom_C(X, F^∗Y) ∼= hom_D(F_∗X, Y). (1.2) Lemma 1.3 Let F:C → D be a covariant functor.

(a) If f: X → Y is a fibration of D-spaces, then F^∗f:F^∗X → F^∗Y is a fibration of C-spaces. If f: X → Y is a cofibration of C-spaces, then F_∗f:F_∗X→F_∗Y is a cofibration of D-spaces;

(b) If X is aC-CW-complex, thenF_∗X is aD-CW-complex;

(c) LetX be a contravariantC-space andEbe a covariantD-spectrum. Then there is a map, natural inX andE,

ΦF :H_q^C(X;F^∗E)→H_q^D(F_∗X;E).

If either

(a) X is aC-CW-complex, or,

(b) for all objectsdofDthe covariantC-setmor_D(d, F(?))is isomorphic to a disjoint union of covariantC-sets of the formmor_C(c0,?)), thenΦ_F :H_q^C(X;F^∗E)→H_q^D(F_∗X;E)is an isomorphism.

Similar statements hold true for cohomology.

Proof: (a) To show that F^∗f is a fibration, one sets up the homotopy lifting problem forC-spaces, solves the adjoint problem for D-spaces, and uses the adjoint property to translate the solution back toC-spaces. The proof for cofibrations is similar.

(b) This follows from two facts. First, for any c ∈ ObC, one can identify F_∗mor_C(?, c) ∼= mor_D(?, F(c)). Second, since F_∗ has a right adjoint, it com- mutes with colimits, in particular, it commutes with pushouts.

(c) Define the map ΦF so that when X⁰ →X is a C-CW-approximation, the following diagram commutes

(F^∗E)^C_q(X⁰) −−−−→^∼= (E)^D_q (F_∗X⁰)

∼=



 y



 y^∼= H_q^C(X⁰;F^∗E) H_q^D(F_∗X⁰;E)

∼=



 y



 y H_q^C(X;F^∗E) −−−−→^ΦF H_q^D(F_∗X;E).

Uniqueness ofC-CW-approximations (up to homotopy) gives that Φ_F is well- defined. If condition (a) holds, then the identity map X → X is a C-CW- approximation and the claim follows.

If condition (b) is satisfied, then a weak homotopy equivalenceX⁰→X gives a weak homotopy equivalenceF_∗X⁰ →F_∗X and hence the lower right vertical map in the above diagram is an isomorphism.

For the applications we are most interested in the most important example is theorbit categoryOr(G) defined for a groupG. The objects are homogeneous G-spacesG/Hand the morphisms are theG-maps. More generally, for a family F of subgroups of G, define the restricted orbit category Or(G,F) to be the category whose objects are the homogeneousG-spacesG/H where H ∈ F and the morphisms are the G-maps. Some examples for F areTR, FIN, VC, and ALL, which are the families consists of the trivial group, the finite subgroups, the virtually cyclic subgroups, and all subgroups respectively.

The remainder of this section is not necessary for the discussion of the p- chain spectral sequence itself in Section 2, but does give the motivation for this paper and is necessary for Section 3. We will review the point of view of [9] con- cerning assembly maps and the Farrell-Jones and Baum-Connes Isomorphism Conjectures.

Definition 1.4 Let F:B → C and E: C → SPECTRA be covariant functors.

Then the composite

H_q^B(?;F^∗E)−−→^ΦF H_q^C(F_∗?;E) ^H

C q(pr;E)

−−−−−−→H_q^C(?;E)

is called the assembly map induced by F, where ΦF is the map appearing in Lemma 1.3(c) andpr :F_∗?→? is the constant map at each object. We some- times abbreviateF^∗E byE.

Remark 1.5 This map can be identified with the map πq(hocolim_BF^∗E)→ πq(hocolim_CE) induced by the functorF.

The three Or(G)-spectra which are useful to us are covariant functors K^alg: Or(G)→SPECTRA

L^hji: Or(G)→SPECTRA K^top: Or(G)→SPECTRA.

These functors were constructed in [9, Section 2], but there was a problem with the construction ofK^topconnected with the pairing on [9, p. 217]. This problem can easily be fixed and the construction can be replaced by more refined ones (see [14]). The key property of these functors is thatπq(K^alg(G/H)) =Kq(RH) for a fixed ringR,πq(L^h−∞i(G/H)) =Lq(RH) for a fixed ringR with involution, and πq(K^top(G/H)) =Kq(C_r^∗H). HereKq(C_r^∗H) is the K-theory of the real or complex reducedC^∗-algebra ofH. In connection with L-groups we use the involution onRGsendingr·gtor·g⁻¹The indexj ∈Z∪ −∞on theL-theory is theK-theory decoration; the important cases for us arej=−∞which arises in the isomorphism conjecture, and the casej= 2 which is used in Wall’s book [32] to classify manifolds. The Isomorphism Conjecture of Farrell and Jones is not true for the decorationsj = 0,1,2, which correspond to the decorations p, handsappearing in the literature [13].

The Isomorphism Conjecture of Baum-Connes for a group G is that the assembly map associated to the inclusion functorI: Or(G,FIN)→Or(G) (see Definition 1.4) yields an isomorphism

H_q^Or(G,FIN)(?;K^top)→H_q^Or(G)(?;K^top) =Kq(C_r^∗(G)).

The Isomorphism Conjecture of Farrell-Jones forRG says that H_q^Or(G,VC)(?;K^alg)→H_q^Or(G)(?;K^alg) =Kq(RG) and

H_q^Or(G,VC)(?;L^h−∞i)→H_q^Or(G)(?;L^h−∞i) =L^h−∞i_q (RG)

are isomorphisms. We will mainly consider the caseR=Z. The point of these conjectures is that they express the target, which is the group one wants to compute, by the source, which only involves theK-theory of the family of finite or virtually cyclic subgroups and is much easier to compute. In the case of the family FIN the source can rationally be computed by equivariant Chern characters [18], [19]. The p-chain spectral sequence is an important tool for integral computations which are much harder. More information about these conjectures can be found for instance in [2], [12], [22] and [31].

2 The p-Chain Spectral Sequence

We establish a spectral sequence converging to the homology (respectively cohomology) of aC-spaceX with coefficients in aC-spectrum E. In the special case whereX =?, this gives a spectral sequence converging to π_∗(hocolim_CE) (respectivelyπ_∗(holim_CE)). It is different from the standard spectral sequence [9, Theorem 4.7], [26, Theorem 8.7], [6, XII, 5.7 on page 339 and XI, 7.1 on page 309] which is an Atiyah-Hirzebruch type spectral sequence and comes from a skeletal filtration. We will need some preliminaries for its construction.

For every non-negative integer p, define the category [p] whose objects are {0,1,2, . . . , p}, with precisely one morphism fromitojifi≤jand no morphism otherwise. Let ∆ be the category of finite ordered sets, i.e. objects are the categories [0],[1],[2], . . .and morphisms are the functors from [p] to [q]. In other words, the morphisms from an object{0,1,2, . . . , p}to an object{0,1,2, . . . , q} are the monotone increasing functions. A simplicial setis a contravariant ∆- set. There is a covariant∆-space ∆_• which sends an object [p] to the standard p-simplex. Thegeometric realization of a simplicial setX_• is the space |X_•|= X_•⊗∆∆_•. Recall that thenerve of a categoryC is the simplicial set

N_pC = functor([p],C)

and its classifying spaceBCis the geometric realization|N_•C|of its nerve. Next we introduce a similar construction. Denote byfunctor([p],^ C) the equivalence classes of covariant functors from [p] to C, where two such functors are called equivalent if they are related by a natural transformation whose evaluation at any object is an isomorphism. More explicitly, functor([p],C) consists of the set of diagrams inC of the shape

c₀ −→^φ0 c₁ −→^φ1 c₂ −→ · · ·^φ2 −−−→^φp−1 c_p

and functor([p],^ C) is the set of equivalence classes under the following equiv- alence relation on functor([p],C): Two such diagrams (c_∗, φ_∗) and (c⁰_∗, φ⁰_∗) are equivalent if there is a commutative diagram with isomorphisms as vertical maps

c0 φ0

−−−−→ c1 φ1

−−−−→ · · · −−−−→^φp−1 cp

∼=





y ^∼=





y ^∼=



 y c⁰₀ ^φ

0

−−−−→0 c⁰₁ ^φ

0

−−−−→ · · ·1 ^φ 0

−−−−→p−1 c⁰_p DefineNe_•C to be the simplicial set given by

NepC = functor([p],^ C)

and BeC to be its geometric realization. We shall proceed to develop basic properties ofBeC analogous to those ofBC, as discussed in [9, pp. 227-229].

If C is a groupoid, i.e. all morphisms are isomorphisms, then BeC is the (discrete) set of isomorphism classes of objects. IfCis a category such that the identity morphisms are the only isomorphisms inC, thenBCis the same asBeC. For instance,B[1] =e B[1] = [0,1].

Lemma 2.1 The projections from C0× C1 toCi for i = 0,1 induce a homeo- morphism

B(e C0× C1) →BeC0×BeC1.

Proof: Given two simplicial sets A_• and B_•, define their productA_•×B_• by sending [p] to the product A_p×B_p. The projections induce a homeomorphism [16, page 43]

|A_•×B_•| → |A_•| × |B_•|.

Now the claim follows since the projections induce isomorphisms of simplicial sets

functor([p],^ C0× C1) → functor([p],^ C0)×functor([p],^ C1).

Given two objects ? and ?? inC, define the category ?↓ C ↓?? as follows: An object is a diagram

?−→^α c−→^β ??

inC. A morphism from ?−→^α c−→^β ?? to ?−→^α0 c⁰−→^β0 ?? is a commutative diagram inC of the shape

? −−−−→^α c −−−−→^β ??

id





y ^φ





y ^id



 y

? ^α

0

−−−−→ c^{0 β}

0

−−−−→ ??

LetBe_p( ?↓ C ↓?? ) be thep-skeleton ofB( ?e ↓ C ↓?? ). We will regardB( ?e ↓ C ↓?? ) as a contravariantC × C^op-space where ? is the variable inC and ?? the variable inC^op. Since maps on classifying spaces induced by functors are cellular, we get a filtration of the contravariantC × C^op-spaceB( ?e ↓ C ↓?? ) by the contravariant C × C^op-spacesBep( ?↓ C ↓?? ) such that

B( ?e ↓ C ↓?? ) = colim

p→∞Be_p( ?↓ C ↓?? ).

Let mor(?,??) be the category whose set of objects is mor(?,??) and whose only morphisms are the identity morphisms of objects. Consider the functor

pr : ?↓ C ↓?? →mor(?,??)

?−→^α c−→^β ??

7→(β◦α: ?→??). It induces a map of contravariantC × C^op-spaces

B(pr) :e B( ?e ↓ C ↓?? ) →B(mor(?,e ??)) = mor(?,??).

LetX be a contravariantC-space. We obtain contravariantC-spacesX⊗_CB( ?e ↓ C ↓?? ) andX⊗Cmor(?,??) where the tensor product is taken over the variable

??. Define a map of contravariantC-spaces

p:X⊗_CB( ?e ↓ C ↓?? ) −−−−−−−→^id⊗CB(pr)e X⊗_Cmor(?,??) −→^∼= X, (2.2) where the second map is the canonical isomorphism given byx⊗φ7→X(φ)(x).

Let n.d.Nep( ?↓ C ↓?? ) denote the set of non-degenerate p-simplices of the simplicial setNe_•( ?↓ C ↓?? ). Elements are given by classes of diagrams

?−→^α c0 φ₀

−→ c1 φ₁

−→ c2 φ₂

−→ · · · −−−→^φp−1 cp

−→β ??

such that noφi is an isomorphism.

Lemma 2.3 (a) We have

X⊗_CB( ?e ↓ C ↓?? ) = colim

p→∞X⊗_CBe_p( ?↓ C ↓?? ) as contravariantC-spaces;

(b) There is a pushout of contravariantC-spaces whose vertical maps are(p− 1)-connected cofibrations of contravariant C-spaces

X⊗_Cn.d.Nep( ?↓ C ↓?? )

×S^p−1→X⊗_C Bep−1( ?↓ C ↓?? )

id⊗Cinc





y ^id⊗Cinc



 y

X⊗_Cn.d.Nep( ?↓ C ↓?? )

×D^p → X⊗_CBep( ?↓ C ↓?? ;

(c) The map p:X⊗CB( ?e ↓ C ↓?? )→X defined in (2.2)is a weak homotopy equivalence of contravariantC-spaces.

Proof: (a) Since the functor X⊗_C−from the category of covariant C-spaces to the category of spaces has a right adjoint, it is compatible with colimits.

(b) There is a canonical CW-structure on the geometric realization of a sim- plicial set whose cells are in bijective correspondence with the non-degenerate simplices [16, page 39]. Hence we get the following pushout of contravariant C × C^op-spaces

n.d.Nep( ?↓ C ↓?? )×S^p−1→Bep−1( ?↓ C ↓?? )



 y



 y n.d.Nep( ?↓ C ↓?? )×D^p → Bep( ?↓ C ↓?? ).

Since the functor X⊗_C −has a right adjoint, it is compatible with pushouts.

For any spaceZ there is a natural homeomorphism

X⊗Cn.d.Nep( ?↓ C ↓?? )

×Z→X⊗C

n.d.Nep( ?↓ C ↓?? )×Z

This shows that the diagram appearing in assertion (b) is a pushout of con- travariant C-spaces. As the inclusion of S^p−1 into D^p is a (p−1)-connected cofibration of spaces, the left vertical and hence also the right vertical arrow are (p−1)-connected cofibrations of contravariantC-spaces.

(c) Fix an objectcin C. Define a functor j⁰: mor(c,??)→ c↓ C ↓??

c−→^α??

7→

c−→^id c−→^α ??

. It induces a map of spaces by

j:X(c)−→^∼= X⊗_CB(mor(c,e ??)) ^idX⊗CBje

0

−−−−−−−→X⊗_CB(e c↓ C ↓?? ).

Define a natural transformation

S: j⁰◦pr(c,??)→id_c↓C↓??

by assigning to an objectc−→^α d−→^β ?? in c↓ C ↓?? the morphism in c↓ C ↓??

c −−−−→^id c −−−−→^β◦α ??

id





y ^α





y ^id



 y c −−−−→^α d −−−−→^β ??

Thus we have a homotopy of maps of covariantC-spaces

h⁰:B(e c↓ C ↓?? )×[0,1]→B(e c↓ C ↓?? )×B[1]e →B(e c↓ C ↓??×[1])→B(e c↓ C ↓?? ) where the first map comes from the identification [0,1] =B[1], the second frome the homeomorphism of Lemma 2.1 and the third from interpretingSas a functor

S: c↓ C ↓??×[1]→ c↓ C ↓??. It induces a homotopy of maps of spaces

h= idX⊗Ch⁰:X⊗CB(e c↓ C ↓?? )×[0,1]→X⊗C B(e c↓ C ↓?? ).

One easily checks thatp(c)◦jis the identity onX(c) andhis a homotopy from j◦p(c) to the identity onX⊗_CB(e c↓ C ↓?? ). Hence

p(c) :X⊗CB(e c↓ C ↓?? )→X(c)

is a homotopy equivalence and in particular a weak homotopy equivalence for all objectsc. This finishes the proof of Lemma 2.3.

Remark 2.4 Notice that the map p:X ⊗_C B( ?e ↓ C ↓?? )→X is not a homo- topy equivalence of contravariantC-spaces. In the proof of Lemma 2.3 we have constructed a homotopy inverse and a corresponding homotopy forp(c) for each objectc, but they do not fit together to an homotopy inverse ofpas a map of contravariantC-spaces. Therefore it is important that we use the (co-)homology H(X;E) which satisfies the WHE-axiom.

We can now apply [9, Theorem 4.7] to the filtration ofX⊗_CB( ?e ↓ C ↓?? ) by the subspacesX⊗CBep( ?↓ C ↓?? ) and obtain:

Theorem 2.5 LetX be a contravariantC-space and Ea covariant respectively contravariantC-spectrum. LetH_p^C(X;E)respectivelyH_C^p(X;E)be the associated homology respectively cohomology theories satisfying the WHE-axiom.

(a) There is a spectral (homology) sequence(E_p,q^r , d^r_p,q)whoseE¹-term is given by

E¹_p,q = H_q^C(X⊗_Cn.d.Nep( ?↓ C ↓?? );E).

The first differential

d¹_p,q: H_q^C(X⊗_Cn.d.Nep( ?↓ C ↓?? );E) → H_q^C(X⊗_Cn.d.Nep−1( ?↓ C ↓?? );E) isPp

i=0(−1)ⁱ·H_q^C(id⊗Cd^p_i;E)whered^p_i:n.d.Nep( ?↓ C ↓?? )→n.d.Nep−1( ?↓ C ↓?? )is thei-th face map. This spectral sequence converges toH_p+q^C (X;E), i.e. there is an ascending filtration F_p,m−pH_m^C(X,E) of H_m^C(X,E) such that

Fp,qH_p+q^C (X,E)/Fp−1,q+1H_p+q^C (X,E) ∼= E_p,q^∞;

(b) Assume that E is a C-Ω-spectrum, i.e. for each object c the spectrum E(c) is an Ω-spectrum. Then there is a spectral (cohomology) sequence (E_r^p,q, d^p,q_r )whoseE¹-term is given by

E₁^p,q = H_C^q(X⊗Cn.d.Nep( ?↓ C ↓?? );E).

The first differential

d^p,q₁ : H_C^q(X⊗_Cn.d.Ne_p( ?↓ C ↓?? );E) → H_C^q(X⊗_Cn.d.Ne_p+1( ?↓ C ↓?? );E) isPp+1

i=0(−1)ⁱ·H^q(id⊗Cd^p_i;E). If one of the following conditions is satis- fied:

(a) The filtration is finite, i.e. there is an integer n >0 such that for any diagram

c0 φ₀

−→c1 φ₁

−→ · · ·−−−→^φn−1 cn

one of the morphismsφi is an isomorphism;

(b) There is n∈Zsuch that πq(E(c))vanishes for all objectsc∈Ob(C) andq < n;

then the spectral sequence converges to H_C^p+q(X;E), i.e. there is a de- scending filtration F^p,m−pH_C^m(X,E) ofH_C^m(X,E)such that

F^p,qH_C^p+q(X;E)/F^p+1,q−1H_C^p+q(X;E) ∼= E_∞^p,q.

Next we want to analyse theE¹-term further. Let Is(C) be the set of isomor- phism classescof objectscinC. Fix for any isomorphism classca representative c∈c. For two objectscanddlet mor_6∼=(c, d) be the subset of mor(c, d) consisting of all morphisms fromctodwhich are not isomorphisms. For an element

c_∗= (c0, c1, . . . , cp)∈

p

Y

i=0

Is(C) andp≥1 define a left-aut(c_p)-right-aut(c₀)-set

S(c_∗) = mor_6∼=(cp−1, cp)×aut(c_p−1)mor_6∼=(cp−2, cp−1)×aut(c_p−2)

. . .×aut(c₁)mor_6∼=(c₀, c₁). (2.6) IfAis a right-aut(cp)-set andBis a left-aut(c0)-set, defineA×aut(cp)S(c_∗)×aut(c0)

Bin the obvious way forp≥1 and byA×aut(c0)Bforp= 0. One easily checks that the map

a

c_∗∈Q_p

i=0Is(C), S(c_∗)6=∅

mor(cp,??)×aut(c_p)S(c_∗)×aut(c₀)mor(?, c0) → n.d.Nep( ?↓ C ↓?? )

which sends the element represented by

(φp, φp−1, . . . , φ0, φ)∈mor(cp,??)×mor_6∼=(cp−1, cp)×. . .×mor_6∼=(c0, c1)×mor(?, c0) to the class of

?−→^φ c₀−→^φ0 c₁−→ · · ·^φ1 −−−→^φp−1 c_p−→^φp ??

is natural in ? and ?? and bijective. Since there is a natural isomorphism of contravariantC-spaces

X⊗_Cmor(c_p,??)×aut(c_p)S(c_∗)×aut(c₀)mor(?, c₀)

∼=

−→ X(c_p)×aut(c_p)S(c_∗)×aut(c₀)mor(?, c₀) we conclude:

Lemma 2.7 There are identifications for theE¹-terms of the spectral sequences in Theorem 2.5

E_p,q¹ = M

c_∗∈Q_p

i=0Is(C), S(c_∗)6=∅

H_q^C(X(cp)×aut(cp)S(c_∗)×aut(c0)mor(?, c0);E)

and

E₁^p,q = M

c_∗∈Q_p

i=0Is(C), S(c_∗)6=∅

H_C^q(X(cp)×aut(cp)S(c_∗)×aut(c0)mor(?, c0);E).

If the category satisfies an additional condition, we can do a much better job identifying theE¹-terms.

Definition 2.8 We callCleft-freeif for any two objectscandc⁰the leftaut(c⁰)- action onmor(c, c⁰)given by composition is free.

For any groupGand any familyFof subgroups the orbit category Or(G,F) is left-free since anyG-map of homogeneousG-spacesG/H→G/Kis surjective.

Let Z be a (left) G-space and let F be spectrum with an action of G by maps of spectra. We can interpretFalso as a covariant Or(G,TR)-spectrum, whereTRis the family consisting of the trivial subgroup ofG. We write

H_q^G(Z;F) := H_q^Or(G,TR)(Z,F). (2.9) Explicitly we get after a choice of a free G-CW-complex Z⁰ together with a G-mapu:Z⁰→Z which is a weak equivalence after forgetting the group action H_q^G(Z;F) = πq(Z₊⁰ ∧GF). (2.10) For instanceH_q^G(?;F) =π_q(EG₊∧GF).

Lemma 2.11 SupposeC is left-free. Then there are identifications for theE¹- terms of the spectral sequences in Theorem 2.5

E_p,q¹ = M

c_∗∈Q_p

i=0Is(C), S(c_∗)6=∅

H_q^aut(c0)(X(cp)×aut(cp)S(c_∗);E(c0))

and

E₁^p,q = M

c_∗∈Q_p

i=0Is(C), S(c_∗)6=∅

H_aut(c^q

0)(X(cp)×aut(c_p)S(c_∗);E(c0)) whereX(c_p)×aut(c_p)S(c_∗)means X(c₀)forp= 0.

With this identification the first differential can be written as

d¹_p,q =

p

X

i=0

(−1)ⁱ(d¹_p,q)i

for certain maps

(d¹_p,q)0:H_q^aut(c0)(X(cp)×aut(cp)S(c_∗);E(c0))

→H_q^aut(c1)(X(cp)×aut(cp)S(c1, . . . , cp);E(c1)),

(d¹_p,q)_i:H_q^aut(c0)(X(c_p)×aut(c_p)S(c_∗);E(c₀))

→H_q^aut(c0)(X(cp)×aut(cp)S(c0, . . . , ci−1, ci+1, . . . , cp);E(c0)),

for0< i < p, and

(d¹_p,q)p:H_q^aut(c0)(X(cp)×aut(c_p)S(c_∗);E(c0))

→H_q^aut(c0)(X(cp−1)×aut(c_p−1)S(c0, . . . , cp−1);E(c0)).

For 0 < i≤ p, these maps are induced by maps of aut(c0)-sets given by con- catenation. The description of (d¹_p,q)0 is more difficult, due to the change of group and coefficients. Let c0 → c1 denote the full subcategory of C with ob- jects {c0, c1}. We label the inclusions of categories i: aut(c0) → (c0 → c1), j: aut(c₁)→(c₀→c₁), andk: (c₀→c₁)→ C. LetY be theaut(c₁)-space

X(c_p)×aut(c_p)S(c₁, . . . , c_p).

Then with the above identification ofE¹ the map (d¹_p,q)₀ is given by H_q^aut(c0)(Y ×aut(c1)mor_6∼=(c0, c1);i^∗k^∗E)

→H_q^c0→c1(i_∗(Y ×aut(c₁)mor_6∼=(c0, c1));k^∗E)

→H_q^c0→c1(j_∗Y;k^∗E)−→^∼= H_q^aut(c1)(Y;j^∗k^∗E), where the first map is the map Φ_i from Lemma 1.3 (c), the middle map is given by the map of (c₀ → c₁)-spaces given as the adjoint of the inclusion of aut(c0)-spaces

Y ×aut(c1)mor_6∼=(c0, c1)→Y ×aut(c1)mor(c0, c1) =i^∗j_∗Y, and the last map is(Φj)⁻¹.

Similar statements are valid in cohomology.

Proof: For an object c0 of C, let ic₀: aut(c0) → C be the corresponding inclusion of categories. Then, according to Lemma 2.7, the E¹-term of the p-chain spectral sequence is identified with

E_p,q¹ = M

c_∗∈Q_p

i=0Is(C), S(c_∗)6=∅

H_q^C((i_c0)_∗(X(c_p)×aut(c_p)S(c_∗));E)

The maps Φi_c0 send the sum below to the sum above to M

c_∗∈Q_p

i=0Is(C), S(c_∗)6=∅

H_q^aut(c0)(X(cp)×aut(cp)S(c_∗);E(c0))

This map between sums is an isomorphism by Lemma 1.3 (c), sinceCis left-free.

We have thus established the first identification in the lemma. The identifica- tions of the differentials are established similarly.

Remark 2.12 Lemma 2.11 shows for a left-free categoryC what the spectral sequence does. Namely, it reduces the computation of the (co-)homology groups of spaces and spectra over a category to the special case of spaces and spectra over a group. The most important case of thep-chain spectral sequence is where X=?is the constant functor given by a point at every object. In this case the aut(c₀)-sets are all discrete and hence a disjoint union of homogeneous spaces aut(c₀)/H, and so the differentials (d¹_p,q)_i for 0 < i≤ p all involve change of group mapsBH₀→BH₁. The remaining differential (d¹_p,q)₀is more subtle and should be thought of as some sort of assembly map. We will comment further on this map in the next section.

Remark 2.13 We have put some effort into avoiding the assumption that C is anEI-category, i.e that all endomorphisms are isomorphisms. Otherwise we would have excluded the orbit category Or(G,F) for F the family of virtually cyclic subgroups of G (see [17, Example 1.32]). But this category appears in the Isomorphism Conjecture in algebraicK or L-theory of Farrell-Jones. The Baum-Connes Conjecture, however, uses only the orbit category Or(G,FIN) forFIN the family of finite subgroups of Gwhich is an EI-category.

If one has an EI-category, the bookkeeping simplifies a little bit. The EI- property makes it possible to define a partial ordering on Is(C) by

c≤d ⇐⇒mor(c, d)6=∅.

We writec < difc≤dandc6=dholds. Ap-chain c_∗ in Cis a sequence c0< c1< . . . < cp.

Letchp(C) be the set ofp-chains. Now one can replace the index set (

c_∗∈

p

Y

i=0

Is(C)

S(c_∗)6=∅ )

in Lemma 2.7 and Lemma 2.11 bychp(C) and replace in the definition ofS(c_∗) the set mor_6∼=(ci−1, ci) by mor(ci−1, ci).

Example 2.14 LetF be a family of subgroups of the (discrete) groupG. We get a bijection

{(H)|H ∈ F} −→^∼= Is(Or(G,F)) (H)7→G/H

where (H) denotes the conjugacy class of a subgroup H ⊆G. LetNH be the normalizer and WH = NH/H be the Weyl group of H ⊆ G. We obtain a bijection

WH −→^∼= aut(G/H) gH 7→ R_g−1:G/H→G/H

whereR_g−1 mapsg⁰H tog⁰g⁻¹H. For (G/H_i|i= 0,1, . . . p) ∈

p

Y

i=0

Is(Or(G,F)) we get

S(G/H_i|i= 0,1, . . . p) = map_6∼=(G/H_p−1, G/H_p)^G×WH_p−1

. . .×WH1map_6∼=(G/H₀, G/H₁)^G, where map_6∼=(G/Hi−1, G/Hi)^G is the set ofG-maps which are not bijective.

Suppose that Or(G,F) is an EI-category. Then a p-chain G/H0 < . . . <

G/Hp is the same as a sequence of conjugacy classes of subgroups (H0)< . . . <(Hp)

where (H_i−1)<(H_i) means thatH_i−1is subconjugated, but not conjugated to (Hi). TheWHp-WH0-set associated to such ap-chain is

S((H0)< . . . <(Hp)) = map(G/Hp−1, G/Hp)^G×WH_p−1

. . .×WH1map(G/H₀, G/H₁)^G. TheE¹-terms of the spectral sequences in Theorem 2.5 become

E_p,q¹ = M

(H₀)<...<(H_p)

H_q^WH0(X(G/Hp)×WHpS((H0)< . . . <(Hp));E(G/H0)) and

E₁^p,q = Y

(H₀)<...<(H_p)

H_WH^q

0(X(G/Hp)×WH_pS((H0)< . . . <(Hp));E(G/H0)) whereX(G/Hp)×WH_pS((H0)< . . . <(Hp)) means X(G/H0) forp= 0.

There is a module or chain complex version of the spectral sequence above.

In the sequel we use the notation and language of [17]. Let R be a commu- tative associative ring with unit andC be a small category. One replaces the contravariant space X by a contravariant RC-chain complex C which satisfies C_p = 0 for p < 0 and the covariant respectively contravariant spectrum E by a covariant respectively contravariantRC-chain complexD which may have non-trivial chain modules in negative dimensions. The role of contravariantC- CW-complexes is now played by projective contravariantRC-chain complexes.

The tensor product ⊗_C is replaced by the tensor product ⊗RC ofRC-modules and the mapping space is replaced by theR-module of homomorphisms ofRC- modules homRC. The homology H_p^C(X;E) now becomes Tor^R_p^C(C, D) and the cohomology H_C^p(X;E) now becomes Ext^p_RC(C, D). Notice that a RC-module can be interpreted as a chain complex concentrated in dimension 0. The proof of the next result is analogous to the results of this section and generalizes the spectral sequence in [17, section 17].

Theorem 2.15 Let M be a contravariant RC-module and N be a covariant respectively contravariantRC-module. Suppose that C is left-free. Then:

(a) There is a spectral (homology) sequence(E_p,q^r , d^r_p,q)whoseE¹-term is given by

E¹_p,q = M

c_∗∈Q_p

i=0Is(C)

Tor^R[aut(c_q ^0)](M(cp)×R[aut(c_p)]RS(c_∗), N(c0))

where R[aut(ci)] is the group ring of aut(ci) with coefficients in R and RS(c_∗)is the free R-module generated by the setS(c_∗). The spectral se- quence converges toTor^R_p+q^C (M, N);

(b) There is a spectral (cohomology) sequence (E^p,q_r , d^p,q_r ) whose E¹-term is given by

E^p,q₁ = Y

c_∗∈Q_p

i=0Is(C)

Ext^q_R[aut(c

0)](M(cp)×R[aut(cp)]RS(c_∗);N(c0)).

The spectral sequence converges toExt^p+q_RC (M, N).

3 Assembly Maps

LetF:B → C andE:C →SPECTRAbe covariant functors. We introduced their assembly map

H_q^B(?;F^∗E)−−→^ΦF H_q^C(?;E)

in Definition 1.4. Recall that we sometimes writeEinstead ofF^∗Eto simplify notation.

LetF:B → C be a covariant functor. For any object c∈ObC, define the undercategoryc ↓F and the overcategoryF ↓c as follows. An object ofc↓ F is a pair (b, φ:c →F(b)) whereb is an object in Band φa morphism in C. A morphismf from (b, φ:c→F(b)) to (b⁰, φ⁰: c→F(b⁰)) is a morphismf:b→b⁰ inB satisfyingF(f)◦φ=φ⁰. An object ofF ↓c is a pair (b, φ:F(b)→c). A morphismf from (b, φ:F(b)→c) to (b⁰, φ⁰: F(b⁰)→c) is a morphismf:b→b⁰ inBsatisfyingφ⁰◦F(f) =φ. We denote the under and overcategories associated to the identity functorF :C → C byc↓ CandC ↓c.

A covariant functor F: B → C is cofinal if for every object c of C, the classifying spaceB(c↓F) is contractible.

For a categoryC, letEC be the contravariantC-space given by EC(?) =B(?↓ C).

TheC-mapEC →? is aC-CW-approximation by [9, p. 230].

Theorem 3.1 (Cofinality Theorem) Let F: B → C be a cofinal covariant functor. LetE: C →SPECTRAbe a covariant functor. Then the assembly map

H_q^B(?;F^∗E)→H_q^C(?;E) is an isomorphism.

Proof: The proof can be found in [15, 4.4]. Here is a proof in our language.

Notice that F_∗EB(?) is a C-CW-complex by Lemma 1.3 (b). Since there is a natural isomorphismF_∗EB(?)∼=B(?↓F) andB(?↓F) is contractible for each

? by cofinality, the unique mapF_∗EB(?)→?is aC-CW-approximation. Using Lemma 1.3 (c), we conclude

H_q^B(?;F^∗E) = H_q^B(EB;F^∗E) = H_q^C(F_∗EB;E) = H_q^C(?;E).

For example, letCbe a category with a final objectc0. LetBbe the subcat- egory with single objectc0and only the identity morphism. Then the inclusion functorF:B → C is cofinal, and so Theorem 3.1 shows that

π_q(E(c₀)) =H_q^B(?;F^∗E)∼=H_q^C(?;E),

which is a well-known fact about homotopy colimits. This also follows from the observation that?_C = mor_C(?, c₀) is aC-CW-complex in this case. Note that Or(G) has a final objectG/G.

Given a functorF :B → C and an objectc in C, there is a commutatative square of functors

F ↓c −−−−→ C ↓^Fc c

Pc





y ^Qc



 y B −−−−→^F C where

Pc(b, φ:F(b)→c) =b,

Fc(b, φ:F(b)→c) = (F(b), φ:F(b)→c), Q_c(c⁰, φ:c⁰→c) =c⁰.

Let E be a covariant C-spectrum. Since C ↓ c has a final object, namely (c,id_c), we get an identification

H_q^C↓c(?;Q^∗_cE) =πq(E(c)).